# Darrol Stinton's statement on wing AR(aspect ratio) for minimum induced drag

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#### Pierre1

##### Active Member
In Darrol Stinton's book 'The design of the aeroplane' page 143, it states that the wing AR of aeroplane around 13 to 15 makes induced drag a minimum while glider induced drag is minimum when wing AR is 40 or more. May I ask why there's a difference? It seems to me that the higher the wing AR, the smaller the induced drag.

Well, I'm not speaking about the other aspects, like the wing structure weight. The topic is only about wing AR and induced drag.

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#### BJC

##### Well-Known Member
HBA Supporter
Induced drag is not a function of wing area (an element of AR); it is a function of wing span. AR, a non-dimensional quantity, simply is not relevant to induced drag.

Edit: Be rigorous about maintaining units when describing physical things. How can a number without units directly describe anything physical? Oh, that? Sure, 42.

BJC

#### Pierre1

##### Active Member
Yes. Induce drag is proportional to 1/b^2 for a certain weight, IAS and Oswald efficiency factor.

I suppose Darrol Stinton's statement in his book is about for a certain wing surface area, so larger AR means larger span.

#### tdfsks

##### Well-Known Member
The equation for induced drag of a wing is Cdi = CL^2/(pi x AR x e). So it is an inverse function of AR (aspect ratio). The bigger AR the smaller the induced drag coefficient (Cdi). e is the Oswalds efficiency or sometimes called the spanwise efficiency factor - basically a fudge factor to account for things such as non optimal spanwise loading and wing root drag. CL is the lift coefficient for the condition that you want to calculate induced drag.

However, for a given planform (i.e. area and AR) induced drag is a function of span loading but then that is just another way of saying that induced drag varies with weight for a given wing at a given angle of attack. That is also the same as saying that it is a function of CL.

I cannot answer the original question as to what Stinton was thinking when he wrote that ... we must have different editions because my 1983 Paperback Edition does not have anything about optimal AR on page 143.

Lets see .... Cdi = Di/qS and CL = L/qS = W/qS and AR = b^2/S so ....

Di = W^2 (pi q S e b^2 / S) = W^2 (pi q e b^2)

So for a given weight W and a given dynamic pressure q induced drag Di is basically a function of 1/b^2. So we are all saying the same thing.

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#### jedi

##### Well-Known Member
In Darrol Stinton's book 'The design of the aeroplane' page 143, it states that the wing AR of aeroplane around 13 to 15 makes induced drag a minimum while glider induced drag is minimum when wing AR is 40 or more. May I ask why there's a difference? It seems to me that the higher the wing AR, the smaller the induced drag.

Well, I'm not speaking about the other aspects, like the wing structure weight. The topic is only about wing AR and induced drag.
Please confirm that the quote is "induced drag" (bold print added for emphasis) and not just minimum drag.

#### BBerson

##### Light Plane Philosopher
HBA Supporter
I suppose Darrol Stinton's statement in his book is about for a certain wing surface area, so larger AR means larger span.
Any discussion of induced drag should include all the details such as what speed is under consideration. Otherwise, general statements are worse than useless.

#### Pierre1

##### Active Member
Any discussion of induced drag should include all the details such as what speed is under consideration. Otherwise, general statements are worse than useless.
I've put on the picture of the text

#### cluttonfred

##### Well-Known Member
HBA Supporter
An AR of 6-9 as per Pierre’s quote seems far more realistic for typical light aircraft.

#### Pierre1

##### Active Member
Any discussion of induced drag should include all the details such as what speed is under consideration. Otherwise, general statements are worse than useless.
But I don't have a real application now. I was just confused by the statement of the book.

#### Pierre1

##### Active Member
An AR of 6-9 as per Pierre’s quote seems far more realistic for typical light aircraft.
Yes. For the compromise with wing structure weight. But for induced drag, it states the best AR is around 13 to 15. That confuses me. Read the whole picture text.

#### cluttonfred

##### Well-Known Member
HBA Supporter
So you are referring to the graph? K is the induced drag factor and A is the aspect ratio, I get that, but I don’t understand the difference between the two sets of graphs for aeroplane vs. sailplane. One seems to indicate an optimal AR around 6 and the other 12 for powered aircraft.

#### TFF

##### Well-Known Member
It’s about practical airplanes for the masses. Gliders are not practical. Efficient, beautiful, quiet but not practical. Low AR are interesting, somewhat small, but are not practical.

You have to define the mission. A Piper Cub, 747, and the Concord are all the best airplanes in the world along with many others. They sure are different, but they all have different missions.

You can build a super long wing plane, where are you going to park it? How crazy are you willing to go with structures. Can’t have the wings on a constant flutter watch. Load?

Theoretical has to meet reality somewhere. Is it custom for you or for the public? Going for a world record or just a regular usage plane? Advanced materials or regular or some other odd thing? It’s about mission and that will shape the plane.

#### cluttonfred

##### Well-Known Member
HBA Supporter
I would appreciate it if someone could explain the graph in post no. 9 for me. I am not clear on the difference between "K" and "K'" or why Stinton would suggest that "K'/A" is so important or why "K/A : K'/A" has a clear minimum value for a powered aircraft but appears to decline forever (theoretically) for a glider.

#### BJC

##### Well-Known Member
HBA Supporter
Everyone has his own definitions of “low AR” and “ideal AR” for a sport aircraft, but the majority of the people actually voting are voting for ARs under 5.

BJC

#### Chilton

##### Well-Known Member
Paperback edittion seems to be Page 133.

Stinton defines K as the induced drag factor for the wing alone, and K' as the induced drag factor for the wing with the rest of the aircraft attached.

So K/A or K'/A is induced drag per unit aspect ratio, and if attaching the rest of the aircraft will likely show differences from powered aircraft to sailplanes.

Since the reference to minimums for powered aircraft or sailplanes was refering to K'/A induced drag per unit AR It is likely I believe to be affected by the speed and angle of attack generally used as a sailplane will be generally operating close to the AOA for best life/drag while powered aircraft generally operate much faster.

#### BBerson

##### Light Plane Philosopher
HBA Supporter
But I don't have a real application now. I was just confused by the statement of the book.
Never heard of K factor so I can't help you. I would just ignore it.

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#### Riggerrob

##### Well-Known Member
Dear TFF,
You made some good points.
Competition sailplanes benefit from high aspect ratios (greater than 20:1) because they are built with precise external loft lines. Over the winter, competitors further refine their competition sailplanes micro-balloons and thousands of hours of sanding to perfect the airfoils.
Few factories can afford to devote that much time to production airplanes and most home-builders lack the tools or skills to build that precisely. With hand tools, few home-builders can build precise airfoils less than 3 or 4 foot chord (about 1 metre). The large chord means that RVs have A/Rs around 5 :1 and STOL home-builts have A/Rs closer to 7:1.

Has anyone compiled lists of home-built chords or aspect-ratios?

Cessna limits A/R to 7 or 10 because that is a reasonable compromise between cost of sheet metal construction and fuel burn.
Boeing and Airbus may be able to built precise wings with A/Rs of 30:1, but they devote millions of dollars to precise production jigs and tooling.
Finally, consider that even a single insect can ruin laminar flow on the best-built wing.

#### rdj

##### Well-Known Member
My paperback 2nd Edition defines K and K' on page 128. He defines K as the 'induced drag factor' and basically 1/e, where e is the Oswald efficiency factor. He states "Generally speaking e = 0.7 to 0.8, making the induced drag factor K = 1.2 to 1.4 for many light aeroplanes." He then states that the interference drag of fuselage/wing junctions affect K, and also "The value of K is also dependent upon the shape of the drag polar, as we shall see in Fig 5.7, and this is a further reason for resorting to use of K' for the applied case, while reserving K for the theoretical wing alone."

Missing from post #9 is the text to the right of the figure, which states "a) Typical values of induced drag factor K and K/A. Actual values for real wings with junctions and other interference effects are K' and K'/A respectively." Combine that with the text in post #7, and it appears to me that figure 4.10a) (post #9) is simply an empirical graph of measured 'induced drag factors' from various aircraft. (Who knows where he got this empirical data from; the text doesn't say.) In any case, if theoretical K is 1.2 to 1.4 and K' factors in wing/body drag et.al., then figure 4.10 a) seems to indicate that "gliders" have very little wing/body interference drag while "aeroplanes" are considerably worse in that department, for some unstated definition of "glider" and "aeroplane".

Too much missing information to interpret figure 4.10 a) quantitatively. I'm inclined to agree with BBerson on this one: "just ignore it".

BJC